Pendidikan Profesi Guru
MODUL
PENDIDIKAN MATEMATIKA
Fraction
for
Junior High School
By
Marsigit
KEMENTERIAN PENDIDIKAN NASIONAL
UNIVERSITAS NEGERI YOGYAKARTA
Program Studi Pendidikan Matematika
2010
0
Preface
The author has been lucky to have a chance to write this Modul. This Modul means
to facilitate the candidate of teachers to learn deeply about the concepts of fraction and its
teaching. It has supporting factors both for the teachers and the students of Junior High
School in their effort to achieve the competences of mathematics.
This module is supporting teaching resources for teaching:
Psikologi Pembelajaran Matematika Program PPG
The author wish to express thank to whoever support this publication. Specifically,
the author would like to give special thank to :
1. Rector of UNY
2. Head of P3AI Yogyakarta State University
3. Dean of the FMIPA
4. Head of Department of Mathematics Education
5. Head of Study Program of Mathematics Education
Hope that by publication of this Modul, the PPG program will be carried out
smoothly.
Yogyakarta, Desember 2010
Head of Departmen of
Mathematics Education
Author
Dr. Hartono
Dr Marsigit, MA
1
TABLE OF CONTENT
Page
Preface
1
Table of Content
2
Introduction
3
Chapter I: The Concept of Fraction
7
Chapter II: Comparing and Ordering Fractions
18
Chapter III: Decimals
27
Chapter IV: Percent and Per mill
31
Chapter V: Working with Fractions
37
Chapter VI: Scientific Notation of Fraction and Rounding-up the Decimals
52
2
Reference
56
Test
57
INTRODUCTION
The new paradigm of teaching needs the teachers to facilitate their students to
develop their competences in mathematics. The teachers also need to develop their
understanding of the nature of school mathematics and the nature the students learn
mathematics. The nature of school mathematics let the students to learn patterns and
relationships, develop mathematical investigations, solve problems and communicate their
mathematics knowledge. In learning mathematics the students need to have a good
motivation and collaboration with their mates. The teacher should facilitate individual
student and give the chance for their students to learn mathematics contextually.
In general, the purpose of publishing this book is to support the government effort in
the the expansion of education, the improvement of quality and relevance; and the
improvement of efficiency and effectiveness. Many efforts become meaningfully prominent
when the government launches the compulsory 9-year basic education; in which the
program is aimed at enrolling all children 7-15 age bracket at the basic education
institutions. In addition to still lacking of school facilities and lacking of text books contribute
to low quality of secondary educations.
Specifically, this Modul is organized to improve intelligence, knowledge, personality,
and skill to be independent and continue to higher level of education. Specifically, the
content standard of mathematics for junior high school outlined that of facilitating the
students:
3
1. to comprehend the mathematical concepts, explain the relationship between the
concepts, apply the concepts and solve the problem.
2. to learn pattern and characteristics of mathematics, practice manipulation to
generalize, organize evidences, and explain mathematics ideas.
3. to comprehend mathematical problems, develop mathematical model, figure out and
obtain the solution of problems.
4. to communicate mathematical ideas using symbols, tables, diagrams, and other
media.
5. to have curiosity to pursue mathematics concepts, develop awareness of
mathematics emerges in daily life, and apply mathematics to solve problems.
There are some hints for the teacher or candidate of the teachers to use this modul in
the case of method as the following:
Mathematical World Orientation
In the first start, it is not so easy for the teacher to develop and manipulate the
concrete material as a mathematical world orientation. It seemed that there are some gaps
between teachers’ habit in doing formal mathematics and informal mathematics. Some
teachers seemed uncertain whether initiating to introduce concrete model to their students
or waiting until their students find for themselves. However most of the teachers believed
that mathematical world orientation is important step to offer the students a motive and a
solution strategy.
Material Model
For the material models, the teacher tried to identify the role of visual representation
in setting up the relationship among fraction concepts, its relations and operations. To some
extent the teachers need to manipulate the concrete model in such a way that they
represent their and the students’ knowledge of fractions. However it seemed that it was not
automatically that the teachers find the supporting aspects of transition toward
4
sophisticated mathematics. Most of the teacher understood that there were problems of
intertwining between informal activities and formal mathematics.
Building Mathematical Relationship
In building mathematical relationship, the teachers perceived that the students need
to develop their mathematical attitude as well as mathematical method. It needed for the
teacher to facilitate students’ questions, students’ interactions and students’ activities.
Uncovering the pattern from material model and trying to connect with mathematical
concepts are important aspects. There were possibilities for the students to find out various
mathematical methods. To compare some mathematical method lead the students have a
clear picture of the problems they faced.
Formal Notation
The teachers perceived that the formal notions of fractions, its relations and
operations come up in line with inclination of sharing ideas of fraction concepts through
small group discussion. The students will find their interest when they get a clear
understanding of formal notions of fractions. The teachers believed that the ultimate
achievement of students is that they feel to have the mathematical concepts they found.
More than this, the students will have an important capacity to a more sophisticated solution
for mathematical problems.
Authors
5
Chapter I
The Concept of Fraction
Standard Competency:
Understanding the characteristics of arithmetical operation of numbers and its application to
solve mathematical problems
Basic Competency
Understanding the concepts of fraction, its type, their relation each other as well as their
operations.
Strategy for Teaching Fractions:
1. Ensure that students have mastered the prerequisite skills for the tasks to be learned
as part of your advance organizer presentation, demonstrate the “Big Idea” and
interconnected relationships between division, fractions, decimals, and percents (e.g.,
physically divide a whole into five equal pieces)
2. Introduce skill instruction with succinct and unambiguous demonstration of the task
to be learned (e.g., solve several problems while the student observes)
3. Introduce instruction using concrete materials (i.e., manipulatives) before proceeding
to semi-concrete materials (e.g., pictorial representations) before proceeding to
abstract problems (e.g., numerical representation)
4. Ensure that your teaching examples include sufficient practice opportunities to
produce task mastery
5. Ensure that your teaching examples include variations of all problem types to avoid
students making incorrect generalizations (e.g., that all fractions represent parts of a
whole)
6. Provide systematic instruction on discriminating among different problem types that
is designed to enable students to know which solution to employ (e.g., teach students
to attend to the operation sign, perhaps by circling it, so that they will select the
correct computational algorithm)
7. Provide guided practice (following teacher-led demonstrations) before assigning
independent work (i.e., have teacher and student work several problems together)
What will you learn in this chapter?
•
Introduction of Fraction
•
Fraction Calculation
•
Scientific Form
6
•
The Rounded of Decimal Fraction
Problem 1:
The word ocean comes from Greece okeanos means river. Approximately 71% of earth is
ocean. More than a half of sea has more than 3000 m deep.The total mass of sea water in
the earth is approximately
kg, or 0,023% of earth total mass.
, and 0,023% are fraction form that you will learn in this chapter.
71%, 1,4
A. Fractions
1. Definition
A fraction is number expressed in the form of
a
, where a and b are integers and b ≠ 0, and
b
b is not a factor of a. The number a is called the numerator, and b is called the
denominator.
Problem:
Notice the following picture carefully.
How you can get the fraction from the whole into the part?
Example
1. Which one of the following numbers is fraction?
a.
1
12
b.
3
2
c.
6
3
Solution
7
a.
1
is fraction, because it matches with the definition.
12
b.
3
is fraction, because it matches with the definition.
2
c.
6
is not fraction, because 3 is a factor of 6.
3
2. See the picture below. How many parts are shadowed ?
Solution
The circle is divide into six equal parts.
The shadowed areas of the circle are 2 of 6 parts.
Hence, the shadowed parts is
2
.
6
Note
6 is the product of 1 × 6, 2 × 6, and 6 × 1 . Hence, the factors of 6 are 1, 2, 3, and 6.
Exercise
1. Which one of the following numbers is fraction?
a.
1
7
b.
3
125
c.
250
25
d.
33
11
e.
100
4
2. How many parts are shadowed from the picture below?
3. Write your name completely. What is the proportion of vocal words compare with the
whole words on your name?
2. Type of Fraction
Problem:
Notice the following picture carefully.
Indicate the fractions that you get from the picture.
8
Source: kids-activities-learning-games.com
a. Proper Fraction
Proper fraction is a fraction where numerator is less than denominator. For example
11 23
3
,
, and .
12 47
6
b. Improper fraction
Improper fraction is a fraction where numerator is more than denominator. For example
5 22
314
,
, and
.
3 7
100
c. Mixed fraction
Teacher’s Guide:
Let the students to learn the following mix number 1 ¾ as simple fraction?
In a small group discussion, let the students discuss the meaning of fraction.
9
The teacher may use ICEBERG approach to encourage the students construct their
own concept of mix fraction, as follow:
Mixed fraction is a fraction that consists of integer number a, b, and c such that a
, where
b
b
=a+
c
c
b
2 8
3
is a pure fraction. For example 1 , 5 , 21 .
c
3 11
7
Mixed fraction may be convert to an improper fraction. Otherwise, we can convert improper
fractions to mixed fractions or whole numbers.
As an illustration, consider the following picture carefully:
10
Source: detub.nl
i). To Convert an Improper Fraction to Mixed Fraction
We can convert improper fractions to mixed fractions or whole numbers by dividing the
numerator by the denominator and expressing any remainder as a fraction of the
denominator.
Example
Convert improper fraction
43
to mixed fraction.
7
Solution
Divide 43 by 7 such as follow
6
43
⇒ 7 43
7
42
−
1
Hence,
43
1
=6
7
7
an integer,
pure fraction
11
Exercise
Convert the following improper fractions to mixed fraction.
1.
8
3
2.
15
20
35
49
3.
4.
5.
4
3
3
6
ii). To Convert a Mixed Fraction to Improper Fraction
We can use the following formula to convert a mixed fraction to improper fraction.
a
b (a × c) + b
=
, where c ≠ 0
c
c
Example
Write down thefollowing mixed fractions in the form of improper fraction.
1. 2
1
4
2. 3
2
7
Solution
1. 2
1 (2 × 4) + 1 8 + 1 9
=
=
=
4
4
4
4
2. 3
2 (3 × 7) + 2 21 + 2 23
=
=
=
7
7
7
7
Exercise
Convert the mixed fraction bellow into an improper fraction.
1. 2
1
2
2. 3
1
4
3. 5
1
6
4. 7
12
2
3
5. 10
11
12
3. Equvalent Fractions
Equivalent fractions are the fractions with has the same location on the line-numbers.
See the picture below.
1
2
1 2
is equivalent to , an this may be written as = .
4
8
4 8
1
2
4
1 2 4
is equivalent to
and , an this may be written as = = .
2
4
8
2 4 8
3
6
3 6
is equivalent to , an this may be written as = .
4
8
4 8
We can get equivalent fractions by multiplying, or dividing, the numerator (top) and
denominator (bottom) of a fraction by the same number ≠ 0 .
Example
Determine two equivalent fractions of
3
.
4
Solution
Multiplied
3
by the number ≠ 0 , e.g. 2 or 3
4
3
3× 2 6
=
is equivalent to
4
4× 2 8
3
3× 3 9
=
is equivalent to
4
4 × 3 12
Hence,
3
9
6
is equivalent to
and
.
8
4
12
Exercise
13
1. Write two equivalent fractions of each of the followings
a.
5
8
b.
31
43
2. Fill in the blank with the correct number to make equivalent fractions.
a.
1 ... 3 ... 9
= = =
=
2 4 ... 10 ...
b.
7 14 42
...
=
=
=
... 28 ... 140
Simplifying Fraction
Simplifying fraction involves dividing the top and bottom by their common factors until this is
no longer possible.
A fraction
a
, where b ≠ 0 can be simplified by dividing the numerator and the
b
denominator by GCF (the Greatest Common Factor) of a and b.
Express with your own word to show the following calculation:
Source: green-planet-solar-energy.com
Example
Write
9
in the simplest form.
36
14
Solution
The GCF of 9 and 36 is 9.
9
→ Divide the numerator and the denominator by 9.
36
9
9:9 1
=
=
36 36 : 9 4
Therefore, the simplest form of fraction
9
1
is .
36
4
Exercise
Write down the following fractions in the simplest form.
1.
2
4
2.
2
6
3.
3
9
4.
15
12
5.
30
36
Story Problems
1. "They" are found on Mars, Venus and Earth. As far as is known there are fifty of "them."
If twenty-two are on Mars, and eighteen are on Earth, what fraction of "them" are on
Venus?
2. Ryan and Kyle had been friends for almost 6 years. Now there was friction between
them. There had been some verbal exchanges, but they had not gotten physical yet.
The argument was over who should get the last piece of the cake Kaylee made for her
birthday party. When Kaylee made the cake she used a ratio of half of a cup of sugar to
every two and a third cups of flour. If she used sixteen and a third cups of flour, how
many cups of sugar did she use?
3. Nicky's uncle wants to paint his apartment in Venice, which is five hundred square
meters. He will have to cover one-fifth of that space with a drop cloth to prevent the
paint from getting on the floor. How many square meters of drop cloth will he need?
15
4. Daniel made a basket of fire starters for his father. He made them by dipping pinecones
in wax. The wax was different colors. When he finished he put them in a pretty basket
with a big red bow on it. He made forty-five fire starters. One-fifth of them were red and
the rest of them were green. How many were green?
5. Brittany and her father went to Paulo's Pizzeria for pizza with everything except
anchovies. The pizza was divided into six slices. Brittany's father ate two-thirds of the
pizza. Brittany ate the rest. How many slices of pizza did Brittany eat?
16
Chapter II
Comparing and Ordering Fractions
Standard Competency:
Understanding the characteristics of arithmetical operation of numbers and its application to
solve mathematical problems
Basic Competency:
Comparing and ordering the fractions
Strategy for Teaching Fractions:
1. Ensure that students have mastered the prerequisite skills for the tasks to be learned
as part of your advance organizer presentation, demonstrate the “Big Idea” and
interconnected relationships between division, fractions, decimals, and percents (e.g.,
physically divide a whole into five equal pieces)
2. Introduce skill instruction with succinct and unambiguous demonstration of the task
to be learned (e.g., solve several problems while the student observes)
3. Introduce instruction using concrete materials (i.e., manipulatives) before proceeding
to semi-concrete materials (e.g., pictorial representations) before proceeding to
abstract problems (e.g., numerical representation)
4. Ensure that your teaching examples include sufficient practice opportunities to
produce task mastery
5. Ensure that your teaching examples include variations of all problem types to avoid
students making incorrect generalizations (e.g., that all fractions represent parts of a
whole)
6. Provide systematic instruction on discriminating among different problem types that
is designed to enable students to know which solution to employ (e.g., teach students
to attend to the operation sign, perhaps by circling it, so that they will select the
correct computational algorithm)
7. Provide guided practice (following teacher-led demonstrations) before assigning
independent work (i.e., have teacher and student work several problems together)
17
What will you learn in this chapter?
•
Comparing the Fractions
•
Ordering the Fractions
Teacher’s Guide:
Compare the following fraction : 2/3 and ½
Which is the bigger?
In a small group discussion, let the students discuss how to compare 2/3 and ½
The teacher may use ICEBERG approach to encourage the students construct their
own ways, as follow:
Proper fractions are located in between 0 and 1 of the line-numbers. Following is the
method to get fractions from line-numbers.
1. Divide the segment line of 0 to 1 in three equal parts, and mark on it the fractions
2
3
2. Compare the fractions 0,
1 2
3
, , and = 1
3 3
3
18
1
and
3
3. You can conclude that 0 <
Thus, the location of
1 2
< < 1.
3 3
2
on the line-numbers is as follow.
3
Example
Mark the location of
5
on the line-numbers
8
Solution
Divide the space between 0 to 1 into eight equal parts
Make a comparison between
You get
1 2 3 4 5 6 7
8
, , , , , , and = 1 .
8 8 8 8 8 8 8
8
1 2 3 4 5 6 7
< < < < < < < 1.
8 8 8 8 8 8 8
Thus, the location of
5
on the line-numbers is as follow.
8
Express with your own word to show the following calculation:
Source: enchantedlearning.com
Exercise
19
Mark on it the following fractions in the line-numbers
1.
3
7
2.
1
9
3.
2
5
4.
3
5
5.
10
11
A. Comparing the Fractions
To compare two fractions with the same denominator, you just need to compare the
numerators.
See the following figure.
You see that the number of shadowed areas on the model of
shadowed area on the model of
Thus,
3
is more than the number of
4
1
.
4
1 3
< .
4 4
Example
Put the sign < or > on
18
3
......
to get correct statement.
7
7
Solution
The fractions have the same denominators, so you just need to compare the numerators.
The numerator of
3
is 3.
7
The numerator of
18
is 18.
7
It’s clear that 3 < 8. So,
3 18
< .
7 7
Express with your own word to describe the following:
20
Source: epotential.education.vic.gov.au
Exercise
Determine the value of fractions indicated by the following diagrams. Then, find out the
greater fraction !
1.
2.
21
To compare two fractions with different denominators, you need to express the fractions in
the same denominators, and then compare the numerator. To have the same
denominators, we need to find the Least Common Multiple (LCM) of the denominators.
Example
Which fraction is greater:
1
3
and ?
5
7
Solution
LCM of 5 and 7 is 35.
1
7
is equivalent to
5
35
3
15
is equivalent to
7
35
We can conclude that
Thus,
7 15
<
.
35 35
1 3
< .
5 7
Note :
LCM is the smallest positive common multiple
Exercise
Fill in the blank the sign of
1.
2 2
...
7 3
2.
3 4
...
9 12
= , < , or >, to compare the following each two fractions.
3.
3 2
...
10 11
4.
13 4
...
15 6
5.
49 21
...
81 27
B. Ordering the Fractions
Deciding the bigger or the smaller of two or more fractions may be called as
ordering the fractions.
22
To order two or more fractions with similar denominators, you just need to order the
numerators. However, if you have fractions with different denominators, how did you decide
what order they should be in? You need first to determine the equivalent fractions.
Source: eduplace.com
Example
Arrange the following fractions in order, with the smallest on the left
Solution
The denominators of the fractions are 5, 10, 15, and 20.
The LCM of 5, 10, 15, and 20 is 60.
You find that :
13
52
is equivalent to
.
15
60
9
54
is equivalent to
.
10
60
23
3
13 9 11
, , , and
5
15 10 20
11
33
is equivalent to
.
20
60
3
36
is equivalent to
.
5
60
Hence, if we put the fractions in order, from the least to the greatest, we get :
and
9
.
10
Exercise
Arrange the following fractions in order, with the smallest on the left
1.
14
11 14
, , and
27
17 16
2.
7
8 6
, , and
8
9 7
3. Express with your own word to show the following calculation:
Source: titirangi.school.nz
24
11 3 13
, , ,
20 5 15
Chapter III
Decimals
Standard Competency:
Understanding the characteristics of arithmetical operation of numbers and its application to
solve mathematical problems
Basic Competency:
Understanding decimals, its type and its characteristics.
Strategy for Teaching Fractions:
1. Ensure that students have mastered the prerequisite skills for the tasks to be
learned
as part of your advance organizer presentation, demonstrate the “Big Idea” and
interconnected relationships between division, fractions, decimals, and percents
(e.g., physically divide a whole into five equal pieces)
2. Introduce skill instruction with succinct and unambiguous demonstration of the task
to be learned (e.g., solve several problems while the student observes)
3. Introduce instruction using concrete materials (i.e., manipulatives) before
proceeding to semi-concrete materials (e.g., pictorial representations) before
proceeding to abstract problems (e.g., numerical representation)
4. Ensure that your teaching examples include sufficient practice opportunities to
produce task mastery
5. Ensure that your teaching examples include variations of all problem types to avoid
students making incorrect generalizations (e.g., that all fractions represent parts of a
whole)
6. Provide systematic instruction on discriminating among different problem types that
is designed to enable students to know which solution to employ (e.g., teach
students to attend to the operation sign, perhaps by circling it, so that they will select
the correct computational algorithm)
7. Provide guided practice (following teacher-led demonstrations) before assigning
independent work (i.e., have teacher and student work several problems together)
What will you learn in this chapter?
•
Understanding decimals
Teacher’s Guide:
Write 0.35 into the simple form of fraction?
25
In a small group discussion, let the students discuss how to write 0.35 into the simple form
of fraction
The teacher may use ICEBERG approach to encourage the students construct their
own ways, as follow:
We can use a decimal point (.) to represent the fraction. Just as places to the left of
the decimal represent units, tens, hundreds, and so on, those to the right of the decimal
represent places for tenths ( ), hundredths ( ), thousandths (
Express with your own word to describe the following:
Source: enchantedlearning.com
26
1
), and so forth.
1000
Example
8
1
= (0 × 1) + (8 × ) , can be written as 0,8
10
10
47
1
1
1
= (0 × 1) + (0 × ) + (4 ×
) + (7 ×
) , can be written as 0,047
100
10
100
1000
2
3
1
1
= (2 × 1) + (0 × ) + (3 ×
) , can be written as 2,03
100
10
100
The fraction 0,8 is written as one decimal point. The fraction 0,047 is written as three
decimal point. And the fraction 2,03 is written as two decimal point.
Example
Write
3
as decimal!
9
Solution
You can write the decimal form of the fraction by dividing the numerator by the
denominator.
0,333...
3
→ 9 3,0
9
2,7 −
30 −
27 −
3
The process of division of 3 by 9 results that
3
= 0,333...
9
27
Exercise
Write the following fractions as decimals:
a. 2
4
25
b.
6
8
You can also represent the decimals into proper fractions. How to do that?
Example
Represent 0,775 into the simplest proper fraction.
Solution
The decimal 0,775 can be written as (0 × 1) + (7 ×
Therefore, 0,775 =
1
1
1
) + (7 ×
) + (5 ×
).
10
100
1000
775
.
1000
The Greatest Common Factor of 775 and 1000 is 25.
Simplify the fraction by dividing the numerator and the denominator by 25.
You get,
775
775 : 25 31
=
=
1000 1000 : 25 40
Hence, the simplest proper fraction form of 0,775 is
28
31
.
40
Exercise
1. Write the following decimals as proper fractions!
a. 5,15 b. 0,124
2. Write the following fractions as three decimal point!
a.
1
7
b.
3
32
29
Chapter IV
Percent and Permill
Standard Competency:
Understanding the characteristics of arithmetical operation of numbers and its application to
solve mathematical problems
Basic Competency:
Understanding percent and permill, their type and their characteristics
Strategy for Teaching Fractions:
1. Ensure that students have mastered the prerequisite skills for the tasks to be
learned
as part of your advance organizer presentation, demonstrate the “Big Idea” and
interconnected relationships between division, fractions, decimals, and percents
(e.g., physically divide a whole into five equal pieces)
2. Introduce skill instruction with succinct and unambiguous demonstration of the task
to be learned (e.g., solve several problems while the student observes)
3. Introduce instruction using concrete materials (i.e., manipulatives) before
proceeding to semi-concrete materials (e.g., pictorial representations) before
proceeding to abstract problems (e.g., numerical representation)
4. Ensure that your teaching examples include sufficient practice opportunities to
produce task mastery
5. Ensure that your teaching examples include variations of all problem types to avoid
students making incorrect generalizations (e.g., that all fractions represent parts of a
whole)
6. Provide systematic instruction on discriminating among different problem types that
is designed to enable students to know which solution to employ (e.g., teach
students to attend to the operation sign, perhaps by circling it, so that they will select
the correct computational algorithm)
7. Provide guided practice (following teacher-led demonstrations) before assigning
independent work (i.e., have teacher and student work several problems together)
What will you learn in this chapter?
•
Percent
•
Permill
30
A. Percent
Teacher’s Guide:
Two third of the Hemisphere consist of water.
State into the percentage the amount of water?
In a small group discussion, let the students discuss how state into the percentage the
amount of water?
The teacher may use ICEBERG approach to encourage the students construct their
own ways, as follow:
Word of percent comes from the word per cent, means per hundred. Percent is a fraction
where the denominator is hundred or per hundred fraction. The notation of percent is %.
a% =
a
, and a % is read a percent.
100
Express with your own word to show the following data:
31
Source: neb-one.gc.ca
Example
5% =
5
1
75 3
= ; 75% =
=
100 20
100 4
Example
1. About
2
of earth surface is covered by water. What is the percentage of the water?
3
Solution
About
2
of earth surface is covered by water.
3
2
200
× 100% =
%.
3
3
You get 66,67 %.
Therefore, 66,67 % of earth surface is covered by water.
2. Aris has two type of books i.e. text book and story books. If Ari’s story books is 40% and
his total books is 120 books, how many Ari’s school books ?
32
Solution
The total books is 120 books, and 40% of total books is story books.
Thus, Ari’s story books is
40
× 120 books = 48 books.
100
Therefore, Ari’s school books is 120 – 48 = 72 books.
Exercise
1. Write the following fractions as percent form!
a.
1
8
b.
13
20
2. Write down the following percentages as simple fractions.
a. 32 %
b. 65 %
3. A 24 meters long hedge has 4 meters been painted. What the percentage of the hedge
has not
been painted yet?
4. There are 70% students of Junior High School present on the Independent Day
celebration. If the total number of students is 40, how many students do not participated in
the celebration?
5. I am the simplest form of a certain fraction. My numerator and denominator are prime
numbers having 2 as their differences. The sum of my numerator and denominator is 12.
What number am I?
B. Per mill
Permil is a fraction where the denominator is thousand or per thousand fraction. The
notation of percent is
0
00
.
33
a 0 00 =
a
, and a
1000
0
00
is read a permil.
Express with your own word to indicate the following symbol:
Source: www.123.rf.com
Example : 5 0 00 =
5
1
150
3
;150 0 00 =
=
=
1000 200
1000 20
Example
1. Write
1
in per thousand.
4
2. Write 750 0 00 in a simple fraction.
Solution
1.
1 1
= × 1000 0 00 = 250 0 00
4 4
2. 750 0 00 =
750
750 : 250 3
=
=
1000 1000 : 250 4
Exercise
1. Write the following fractions into per thousand.
34
a.
1
16
b.
3
20
c. 2
1
4
d.
11
20
e. 1
3
10
2. Write down the following per thousand as a simple fraction.
a. 240 0 00
b. 850 0 00
c. 380 0 00
35
d. 780 0 00
e. 125 0 00
Chapter V
Working with Fractions
Standard Competency:
Understanding the characteristics of arithmetical operation of numbers and its application to
solve mathematical problems
Basic Competency:
Working with fractions i.e. adding, subtracting, multiplying and dividing them
Strategy for Teaching Fractions:
1. Ensure that students have mastered the prerequisite skills for the tasks to be
learned
as part of your advance organizer presentation, demonstrate the “Big Idea” and
interconnected relationships between division, fractions, decimals, and percents
(e.g., physically divide a whole into five equal pieces)
2. Introduce skill instruction with succinct and unambiguous demonstration of the task
to be learned (e.g., solve several problems while the student observes)
3. Introduce instruction using concrete materials (i.e., manipulatives) before
proceeding to semi-concrete materials (e.g., pictorial representations) before
proceeding to abstract problems (e.g., numerical representation)
4. Ensure that your teaching examples include sufficient practice opportunities to
produce task mastery
5. Ensure that your teaching examples include variations of all problem types to avoid
students making incorrect generalizations (e.g., that all fractions represent parts of a
whole)
6. Provide systematic instruction on discriminating among different problem types that
is designed to enable students to know which solution to employ (e.g., teach
students to attend to the operation sign, perhaps by circling it, so that they will select
the correct computational algorithm)
7. Provide guided practice (following teacher-led demonstrations) before assigning
independent work (i.e., have teacher and student work several problems together)
What will you learn in this chapter?
•
Adding Like Fractions
•
Adding Unlike Fractions
•
Subtracting Like Fractions
•
Subtracting Unlike Fractions
36
•
Multiplying the Fractions
•
Dividing the Fractions
A. Adding and Subtracting the Fractions
1. Adding the Fractions
Source: theapple.monster.com
1) Adding the Like Fractions
To add two like fractions, you just need to add the numerator while the denominator
remains unchanged..
a b a+b
+ =
, where c ≠ 0
c c
c
Example
Calculate the following addition of fractions.
1.
3 2
+
8 8
2.
4 ⎛ −3⎞
+⎜
⎟
9 ⎝ 9 ⎠
3. 3
Solution
1.
3 2 3+ 2 5
+ =
=
8 8
8
8
2.
4 ⎛ − 3 ⎞ 4 + (−3) 1
+⎜
=
⎟=
9 ⎝ 9 ⎠
9
9
37
1
3
+5
7
7
3. 3
1
3 22 38 22 + 38 60
4
+5 =
+
=
=
=8
7
7 7
7
7
7
7
Note :
Mixed fraction a
(a × c ) + b .
b
can be written as an improper fraction
c
c
2) Adding Unlike Fractions
To add two unlike fractions, you have to convert the fractions to like fractions first by using
the Least Common Multiple (LCM) of the denominators and then add the numerators.
Example
Calculate the following addition of fractions!
1.
3 ⎛ −1⎞
+⎜ ⎟
4 ⎝ 7 ⎠
2. 1,37 + 2,18
Penyelesaian
1.
3 ⎛ − 1 ⎞ 21 ⎛ − 4 ⎞ 17
+⎜
+⎜ ⎟ =
⎟=
4 ⎝ 7 ⎠ 28 ⎝ 28 ⎠ 28
2. 1,37 + 2,18 =
The LCM of 4 and 7 is 28
123 218 355
1
1
+
=
= (3 × 1) + (5 × ) + (5 ×
) = 3,55
100 100 100
10
100
Exercise
Find the result of each of the following addition.
1. 2
1
2
+ 12
2
7
38
2. 0,03 + 1,2 + 10,08
B. Subtracting the Fractions
Teacher’s Guide:
Agus has bought Tart Cake. He cut it into eight similar parts. He and his three younger
brothers will each take one part of it. And he will save the rest. How many parts of the Cake
Agus will save?
In a small group discussion, let the students discuss how many parts of the Cake Agus will
save?
The teacher may use ICEBERG approach to encourage the students construct their
own ways, as follow:
1. Subtracting Like Fractions
39
Subtracting the fractions is the opposite of adding fractions. To subtract like fractions, you
just need to subtract the numerator while the denominator remain unchanged.
a b a−b
− =
, where c ≠ 0
c c
c
Express with your own word to show the following calculation:
Source: helpwithfractions.com
Example
Calculate the subtraction fractions below.
1.
4 3
−
7 7
2.
6
3
5 ⎛ 1⎞
− ⎜ − ⎟ 3. 5 − 3
9 ⎝ 9⎠
8
8
Solution
1.
4 3 4−3 1
− =
=
7 7
7
7
2.
5 ⎛ 1 ⎞ 5 − (−1) 5 + 1 6
− ⎜− ⎟ =
=
=
9 ⎝ 9⎠
9
9
9
3. 5
6
3 46 27 19
3
−3 =
−
=
=2
8
8 8
8
8
8
Exercise
Find the result of the following subtraction.
1.
2 18
−
27 27
2.
2 ⎛ 24 ⎞
− ⎜− ⎟
52 ⎝ 52 ⎠
40
3. 4
4.
5
6
−2
7
7
15 ⎛ 37 ⎞
− ⎜− ⎟
30 ⎝ 30 ⎠
3
5
5. 5 − 3
1
5
2. Subtracting the Unlike Fractions
To subtract unlike fractions, you have to convert the fractions to like fractions first by using
the Least Common Multiple (LCM) of the denominators and then subtract the numerators.
Example
Calculate the following subtraction fractions.
1. 2
3
5
−1
8
7
2. 6,28 – 0,37
Solution
1. 2
3 5 19 12 133 96 133 − 96 37
−1 =
− =
−
=
=
8 7 8 7
56 56
56
56
The LCM of 8 and 7 is 56.
2. 6,28 − 0,37 =
628 37 591
1⎞ ⎛
1 ⎞
⎛
−
=
= (5 × 1) + ⎜ 9 × ⎟ + ⎜1 ×
⎟ = 5,91
100 100 100
⎝ 10 ⎠ ⎝ 100 ⎠
41
Exercise
1. Find the results of the following subtraction!
a. 2
1
2
− 12
2
7
b. 2,93 – 0,13
2. Mr. Amran had 3,87 litres of gasoline on his motor tank. About 0,19 litres have been
spent by Mr. Amran to go to his office. How much of the gasoline on his motor tank
remained ?
B. Multiplying and Dividing the Fractions
1. Multiplying the Fractions
The multiplication of the fractions
a
c
and , where b ≠ 0 , and d ≠ 0 , is
b
d
a c a×c
× =
, b ≠ 0, d ≠ 0
b d b×d
Here are some properties of multiplication the fractions.
42
1. Commutative
a × b = b × a , where a and b are fractions.
2. Associative
(a × b ) × c = a × (b × c ) , where a , b and c are fractions.
3. Distributive
a × (b + c) = (a × b ) + (a × c ) , where a , b and c are fractions.
Example
Find the results of the following multiplication
1.
7 2
×
12 17
1 ⎛
5 ⎝
2. − 3 × ⎜ − 7
2⎞
⎟
11 ⎠
3. 0,35 x 1,42
Solution
1.
7 2
7×2
14
7
× =
=
=
12 17 12 × 17 204 102
1 ⎛
5 ⎝
2. − 3 × ⎜ − 7
2⎞
16 ⎛ 79 ⎞
16 × (− 79) 1264
=
⎟ = − ×⎜− ⎟ = −
11 ⎠
5 ⎝ 11 ⎠
5 × 11
55
3. 0,35 × 1,42 =
35 142 35 × 142
4970
×
=
=
= 0,497
100 100 100 × 100 10000
Exercise
Calculate the following multiplication fractions.
1.
21 17
×
43 57
4. 2,41 × 0,37
2. 9 ×
3 ⎛
5 ⎝
7
15
1⎞
3⎠
3. 5 × ⎜ − 2 ⎟
⎛
⎝
1⎞
9⎠
5. − 5,17 × ⎜ − 5 ⎟
2. Dividing the Fractions
43
Teacher’s Guide:
Fifteen kilogram of rice will be put into some smaller containers size ¾ kg. How many
containers does it need?
In a small group discussion, let the students discuss how many containers does it need?
The teacher may use ICEBERG approach to encourage the students construct their
own ways, as follow:
The division of fraction
a
c
by
where b ≠ 0 , c ≠ 0 and d ≠ 0 , is
b
d
a c a d
: = × , where b ≠ 0 , c ≠ 0 and d ≠ 0
b d b c
Find the results of the following divisions.
1. 6 :
1
8
2.
1
:6
8
3.
3 1
:
2 5
44
4. 9 : 3
2
5
5. 0,05 : 0,31
Solution
1. 6 :
1 6 1 6 8 6×8
= : = × =
= 48
8 1 8 1 1
1
2.
1
1 1 1
:6 = × =
8
8 6 48
3.
3 1 3 5 15
1
: = × =
=7
2 5 2 1 2
2
4. 9 : 3
2
17 9 5
9 × 5 45
11
=
=2
= 9:
= × =
5
5 1 17 1 × 17 17
17
5. 0,05 : 0,31 =
5 31
5 100 5 × 100
500
500 : 100
5
=
×
=
:
=
=
=
100 100 100 31 100 × 31 3100 3100 : 100 31
Exercise
1. Find the results of the following divisions.
a.
1 1
:
3 4
b.
3
: (− 12)
5
c. 27 : 0,25
d. 3 :
e. 4
1
3
1
: (− 0,28)
7
45
2. The ticket price on “Galaksi” theatre on Tuesday-Monday is Rp 15.000,00. Special offer
on Monday is that
5
of the ticket price on Tuesday-Monday. Pipit brought Rp 50.000,00
6
and she wants with 4 of her friends to see a movie on Monday . Has Pipit enough money to
see the movie with her friends?
3. Mrs. Rosita bought 5 cakes that will be given to her children. Each of her children will get
1
1
cakes. How many children Mrs. Rosita has ?
4
C. Exponent of the Fractions
n
⎛a⎞
The exponent form of the fractions can be written as ⎜ ⎟ .
⎝b⎠
n
a a a
a
⎛a⎞
⎜ ⎟ = × × .... ×
b b b
b
⎝b⎠
a × a × a × ..... × a
b × b × b × ..... × b
an
= n
b
=
Here are some properties of the exponent of the fractions.
m
n
⎛a⎞ ⎛a⎞
⎛a⎞
1. ⎜ ⎟ × ⎜ ⎟ = ⎜ ⎟
⎝b⎠ ⎝b⎠
⎝b⎠
m+ n
=
a m+ n
b m+n
m
⎛a⎞
⎜ ⎟
m−n
a m−n
b⎠
⎛a⎞
⎝
2.
=
=
, where m > n
⎜
⎟
n
b m−n
⎝b⎠
⎛a⎞
⎜ ⎟
⎝b⎠
46
n
m× n
⎛⎛ a ⎞m ⎞
a m× n
⎛a⎞
⎜
⎟
3. ⎜ ⎟
=⎜ ⎟
= m× n
⎜⎝ b ⎠ ⎟
b
⎝b⎠
⎝
⎠
Example
Find the results of the following operations.
3
3
2
⎛1⎞ ⎛1⎞
1. ⎜ ⎟ × ⎜ ⎟
⎝ 3⎠ ⎝ 3⎠
⎛1⎞
⎜ ⎟
⎝2⎠
2.
3
⎛1⎞
⎜ ⎟
⎝2⎠
⎛⎛ 1 ⎞2 ⎞
3. ⎜ ⎜ ⎟ ⎟
⎜⎝ 3 ⎠ ⎟
⎝
⎠
3
Solution
3
2
⎛1⎞ ⎛1⎞
⎛1⎞
1. ⎜ ⎟ × ⎜ ⎟ = ⎜ ⎟
⎝3⎠ ⎝3⎠
⎝3⎠
3+ 2
5
1× 1×1× 1× 1
1
⎛1⎞
=⎜ ⎟ =
=
3 × 3 × 3 × 3 × 3 243
⎝3⎠
3
⎛1⎞
⎜ ⎟
5− 2
3
1× 1× 1 1
2⎠
⎛1⎞
⎛1⎞
⎝
=⎜ ⎟ =⎜ ⎟ =
=
2.
3
2× 2× 2 8
⎝2⎠
⎝2⎠
⎛1⎞
⎜ ⎟
⎝2⎠
3
2×3
6
⎛⎛ 1 ⎞2 ⎞
1× 1× 1× 1× 1× 1
1
⎛1⎞
⎛1⎞
⎜
⎟
3. ⎜ ⎟
=
=⎜ ⎟ =⎜ ⎟ =
⎜⎝ 3 ⎠ ⎟
3 × 3 × 3 × 3 × 3 × 3 729
⎝ 3⎠
⎝3⎠
⎝
⎠
Exercise
Find the result of each of the following.
2
⎛1⎞ ⎛1⎞
1. ⎜ ⎟ × ⎜ ⎟
⎝5⎠ ⎝5⎠
2.
3
(0,3)5
(0,3)2
47
⎡⎛ 1 ⎞ 3 ⎤
3. ⎢⎜ − ⎟ ⎥
⎣⎢⎝ 4 ⎠ ⎦⎥
2
4. Express with your own word to show the following calculation:
48
Chapter VI
Scientific Notation of Fraction and Rounding-up the Decimals
Standard Competency:
Understanding the characteristics of arithmetical operation of numbers and its application to
solve mathematical problems
Basic Competency:
Expressing the scientific notation of fractions
Strategy for Teaching Fractions:
1. Ensure that students have mastered the prerequisite skills for the tasks to be
learned
as part of your advance organizer presentation, demonstrate the “Big Idea” and
interconnected relationships between division, fractions, decimals, and percents
(e.g., physically divide a whole into five equal pieces)
2. Introduce skill instruction with succinct and unambiguous demonstration of the task
to be learned (e.g., solve several problems while the student observes)
3. Introduce instruction using concrete materials (i.e., manipulatives) before
proceeding to semi-concrete materials (e.g., pictorial representations) before
proceeding to abstract problems (e.g., numerical representation)
4. Ensure that your teaching examples include sufficient practice opportunities to
produce task mastery
5. Ensure that your teaching examples include variations of all problem types to avoid
students making incorrect generalizations (e.g., that all fractions represent parts of a
whole)
6. Provide systematic instruction on discriminating among different problem types that
is designed to enable students to know which solution to employ (e.g., teach
students to attend to the operation sign, perhaps by circling it, so that they will select
the correct computational algorithm)
7. Provide guided practice (following teacher-led demonstrations) before assigning
independent work (i.e., have teacher and student work several problems together)
What will you learn from this chapter?
•
Scientific Notation
•
Rounding-up the Decimals
49
A. Scientific Notation
Scientific notation is a short way to write a very large or a very small number to make it
more efficient in its writing.
Here are some rules of scientific notation.
1. For numbers bigger than 10, the scientific notation is
, with
and
is the real number.
2. For numbers between 0 and 1, the scientific notation is
and
, with
is the real number.
Example
Write down the number below in the scientific notation.
1. 2.732
2. 0,000253
Solution :
a.
number of . You have,
2,732
b.
rule with
2.732 is bigger than 10. So, use
= 2,732 and
with the real
= 3.So the scientific notation of 2.732 is
.
0,000253 is the number between 0 and 1. So use
rule with
with the real number of . So the scientific notation of 0,000253 is 2,53
50
.
Exercise
Write down the following number as a scientific form!
1.
278.000
2.
0,000008654
B. Rounding-up the Decimals
The purpose of rounding-up the decimals is to approximate a number by replacing it
with a simpler number, one with fewer digits or one with zeros for its ending digits.
There are two rules in rounding-up the decimals
5 it should be rounded-up.
1.
If the number is
2.
If the number is less than 5 it should not be rounded-up.
Examples
Do the following task!
1.
Round 6,321 off up to two decimal places!
2.
Round 7,461 off up to one decimal place!
Solution :
1.
6,321 has three places of decimal. Because of 1<5 so the round off is 6,32. So the
round off 6,321 until two places of decimal is 6,32.
2.
7,461 has three places of decimal.
•
The last number of 7,461 is 1, Because of 1<5 so the round off is 7,46.
51
•
The last number of 7,46 is 6. Because of 6>5 so the round off is 7,5
So the round off 7,461 until one place of decimal is 7,5
Exercises
Do the following task!
1.
Round 78,3646 off until two places of decimal!.
2.
Round 4,5562396 off until one place of decimal!
References
Emilie A. Naiser ,et al., 2004, Teaching Fractions: Strategies Used for Teaching Fractions to
Middle Grades Students. Journal of Research in Childhood Education, Vol. 18
Ernest, P. (1991). The Philosophy of Mathematics Education. Hampshire: The Falmer
Press.
Freudenthal, H. (1991). Revisiting Mathematics Education. China Lectures. Dordrecht:
Kluwer Academic Publishers.
Gravemeijer, K.P.E. (1994). Developing Realistic Mathematics Education. Utrecht: CD-ß
Sembiring, at al., 2010, A Decade of PMRI in Indonesia, Utrecht: Ten Brink, Meppel
Sutarto Hadi, 2008, RME in Primary School, Power Point Presentation
Zulkardi (2006). How to Design Mathematics Lessons based on the Realistic Approach?
Retreived 2006 <http//www.google.com>
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53